SET-PARTITION TABLEAUX Tom Halverson Macalester College FPSAC 2019 - PowerPoint PPT Presentation

SET-PARTITION TABLEAUX Tom Halverson Macalester College FPSAC 2019 Ljubljana July 2, 2019 1/25

Set-Partition Tableaux Integer Partition: Organization: I. Origins: Representation 8 , 13 14 Theory of the Symmetric 2 , 3 1 , 6 , 10 16 Group 4 12 II. Schur-Weyl Duality: The 5 , 7 , 9 15 , 17 Partition Algebra and Other Diagram Algebras 11 III. Insertion Bijections Set Partition: May 2016: 8 9 { 2 , 3 } , { 4 } , { 5 , 7 , 9 } , [Benkart-H-Harmon] Dimensions of > > > > < = { 1 , 6 , 10 } , { 11 } , { 12 } , irreducible modules ... { 8 , 13 } , { 14 } , { 16 } , [Orellana-Zabrocki] Symmetric > > > > : ; { 15 , 17 } group characters as symmetric functions 1/25

I. Symmetric Group Tensor Power Representations 1/25

Origins: The Symmetric Group S n I M n = n -dimensional permutation module ⇠ = S n � S n I Basis: v 1 , v 2 , . . . , v n with group action: � ( v i ) = v σ ( i ) I S λ n = irreducible C S n -module, � ` n I M ⌦ k = k -fold tensor product module n I Diagonal action on basis of simple tensors: � ( v i 1 ⌦ v i 2 ⌦ · · · ⌦ v i k ) = v σ ( i 1 ) ⌦ v σ ( i 2 ) ⌦ · · · ⌦ v σ ( i k ) Question Determine the multiplicity m λ k , n in the decomposition: M M ⌦ k m λ k , n S λ = n n λ ` n 2/25

Method 1: Restriction-Induction Tensor Identity: tensoring with the permutation module is the same as restriction and induction n ⌦ M n ⇠ S λ = Ind S n S n − 1 Res S n S n − 1 ( S λ n ) M ⇠ = Ind S n S ν n � 1 S n − 1 ν = λ � ⇤ M M ⇠ S µ = n µ = ν + ⇤ ν = λ � ⇤ M n ⌦ M n ⇠ S µ S λ = n µ =( λ � ⇤ )+ ⇤ 3/25

Bratteli Diagram: B ( S 5 , S 4 ) for M ⌦ k 5 M ⊗ 0 5 1 M ⊗ . 5 5 1 M ⊗ 1 5 1 1 M ⊗ 1 . 5 5 1 2 M ⊗ 2 1 1 5 3 2 M ⊗ 2 . 5 1 1 5 5 5 M ⊗ 3 6 6 2 1 5 10 5 M ⊗ 3 . 5 15 8 9 1 5 22 . . . . . . . . . . . . . . . . . . 4/25

Vacillating Tableaux A vacillating tableaux of shape � ` n is a sequence of partitions for which 2 = � i � ⇤ � i + 1 and � i +1 = � i + 1 2 + ⇤ . Example : A vacillating tableaux of length 6 and shape : 0 1 2 3 4 5 6 The multiplicity of S λ n in M ⌦ k is given by n m λ k , n = # vacillating tableaux of length k and shape � . 5/25

Method 2: Decompose M ⌦ k into Permutation Modules n I Diagonal Action: � ( v a ⌦ v a ⌦ v b ⌦ v a ⌦ v b ⌦ v c ⌦ v d ⌦ v c ) v σ ( a ) ⌦ v σ ( a ) ⌦ v σ ( b ) ⌦ v σ ( a ) ⌦ v σ ( b ) ⌦ v σ ( c ) ⌦ v σ ( d ) ⌦ v σ ( c ) = 1 2 3 4 5 6 7 8 I Partition tensor positions: P = { 1 , 2 , 4 | 3 , 5 | 6 , 8 | 7 } : v i j = v i ` i ff j ⇠ ` in P I As a , b , c , d vary over distinct elements of { 1 , . . . , n } , these simple tensors span a submodule isomorphic to the permutation module M ( n � 4 , 1 , 1 , 1 , 1) = Ind S n S n − 4 ⇥ S 1 ⇥ S 1 ⇥ S 1 ⇥ S 1 ( 1 ) . 6/25

Decompose M ⌦ k into Permutation Modules n n n ⇢ k � ⇢ k � M M M M ( n � t , 1 t ) ⇠ ⇠ M ⌦ k K λ , ( n � t , 1 t ) S λ = = n n t t t =0 t =0 λ ` n n ⇢ k � M M ⇠ f λ / ( n � t ) S λ = n t t =0 λ ` n ⇢ k � I = # set partitions of { 1 , . . . , k } into t subsets (Stirling 2 nd ) t I K λ , ( n � t , 1 t ) = Kostka number = #semistandard tableaux of shape � filled with 0 , . . . , 0 , 1 , 2 , . . . , t . | {z } n � t 0 0 0 3 7 1 4 6 8 I Example: has � = (5 , 4 , 2 , 1) , n = 12, t = 9. 2 9 5 I K λ , ( n � t , 1 t ) = f λ / ( n � t ) = # standard tableaux shape � / ( n � t ) 7/25

Decompose M ⌦ k into Permutation Modules n n ⇢ k � M M ⇠ f λ / ( n � t ) S λ M ⌦ k = n n t t =0 λ ` n � n ⇢ k � ⇢ � P = partition of { 1 , . . . , k } into t parts X f λ / ( n � t ) = # � m λ k , n = ( P , T ) � t T = standard tableau of shape � / ( n � t ) � t =0 Example . A standard set-partition tableau of shape � = (5 , 4 , 2 , 1) P = { 1 , 6 | 4 , 7 , 9 , 10 | 2 , 11 , 12 | 8 , 14 | 15 , 16 | 5 , 13 , 18 | 3 , 17 , 19 | 20 } t = 8 . 3 , 17 , 19 1 , 6 4 , 7 , 9 , 10 5 , 13 , 18 20 T = 2 , 11 , 12 8 , 14 n = 12 15 , 16 8/25

Set-Partition Tableaux ! Vacillating Tableaux 8 9 8 9 Standard set-partition Vacillating tableaux < = < = m λ k , n = # tableaux of shape ; = # of length k : : ; � / ( n � k ) and shape � 8 9 ( 7 ) 6 0 1 2 3 4 5 6 < = 4 27 ! : 135 ; 9/25

0 1 2 3 4 5 6 7 6 ! 4 27 135 [BH’19] H-Benkart, [COSSZ’19] Colmenarejo, Orellana, Saliola, Schilling, Zabrocki 6 6 2 2 2 13 4 27 2 13 4 4 4 135 135 135 → 6 6 2 → 27 2 4 → 4 2 13 4 → 135 4 135 135 1. Remove box containing 1 1 m = max ( T ) 2 2. Delete m from the box 3. Schensted insert box → 2 → 1 2 → 13 1 in T > 1 10/25

II. Schur-Weyl Duality and the Partition Algebra 10/25

Centralizer Algebra of S n on M ⌦ k n Centralizer Algebra: n o � � �� ( x ) = �� ( x ) , � 2 S n Z k , n := End S n ( M ⌦ k � 2 End ( M ⌦ k n ) = n ) Schur-Weyl Duality: M M ⇠ ⇠ M ⌦ k m λ k , n S λ f λ Z λ = = n n k , n λ ` n λ ` n | {z } | {z } as an S n -module as a Z k , n -module I m λ k , n = mult k ( S λ n ) = dim( Z λ k , n ) = #(Standard Set-Partition Tableaux) I f λ = dim( S λ n ) = mult( Z λ k , n ) = # (Standard Tableaux) X ( m λ k , n ) 2 Artin-Wedderburn theory: dim( Z k , n ) = λ ` n 11/25

Bratteli Diagram: B ( S 6 , S 5 ) = B ( Z k , 6 ) Sum of Squares (Bell No’s) k = 0 1 1 k = . 5 1 1 k = 1 2 1 1 k = 1 . 5 5 1 2 2 2 + 3 2 + 1 1 + 1 2 = 15 k = 2 1 1 3 2 k = 2 . 5 52 1 1 5 5 k = 3 203 6 6 1 2 1 10 5 k = 3 . 5 876 15 22 9 9 2 1 12/25

Partition Algebra P k ( n ) [P.P. Martin, V.F.R. Jones, ⇡ 1993] Basis of set partitions of { 1 , . . . , k , 1 0 , . . . , k 0 } . 1 2 3 4 5 6 7 8 ⇢ { 1 , 3 , 4 0 } , { 2 , 1 0 } , { 4 , 5 , 7 } , � = { 6 , 8 0 } , { 8 , 6 0 } , { 2 0 , 3 0 } , { 5 0 , 7 0 } 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 Multiplication given by diagram concatenation: = n Generated by 3 types of diagrams: , , 13/25

Action of P k ( n ) on Tensor Space M ⌦ k n I Commutes with S n : Transposition: P k ( n ) ! End S n ( M ⌦ k n ) u 1 u 2 u 3 u 4 u 5 u 6 v 7 v 8 I is surjective I it is injective if n � 2 k u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 the stable case I kernel? [Benkart-H’19] u 1 u 2 u 3 u 4 u 6 v 7 v 8 v n X v = v i projection onto trivial module u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 i =1 δ u 4 , u 5 u 1 u 2 u 3 u 4 u 4 u 6 v 7 v 8 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 14/25

Irreducible Modules for the Partition Algebra M M ⇠ ⇠ Schur-Weyl Duality: M ⌦ k m λ k , n S λ f λ P λ = = n n k λ ` n λ ` n | {z } | {z } as an S n -module as a P k ( n )-module I m λ k , n = dim( P λ k ) = #(Standard Set-Partition Tableaux) I f λ = dim( S λ n ) = # (Standard Tableaux) The irreducible partition algebra module: ⇢ ⇢ standard set-partition � � � P λ � k = C -span � T 2 v T tableaux of shape � Question Is there a combinatorial action analogous to Young’s repre- sentations of S n on standard tableaux? 15/25

Action on Basis Indexed by Set-Partition Tableaux [H-Jacobson, 2018] 1 2 3 4 4 5 6 7 8 9 10 11 12 13 10 13 4 10 , 13 · · · 3 , 5 , 6 11 · · · = 1 , 2 , 3 8 , 11 9 1 , 2 4 12 8 , 9 , 10 7 , 13 5 , 6 , 7 12 1 2 3 4 5 6 7 8 9 10 11 12 13 · · · 3 , 5 , 6 11 · · · 5 8 , 9 10 , 12 , 13 1 , 2 4 12 = 1 , 2 , 4 3 , 6 = 0 8 , 9 , 10 7 , 13 1 , 2 , 4 7 , 11 16/25

Action of Generators , , 1 2 3 4 5 6 7 8 9 3 , 5 1 , 3 · · · · · · 1 , 2 = 2 , 4 4 6 5 7 7 , 9 8 , 9 8 6 17/25

Action of Generators , , 1 2 3 4 5 6 7 8 9 · · · 2 3 , 5 3 , 5 · · · = 1 4 6 1 , 2 4 6 7 , 9 7 , 9 8 8 1 2 3 4 5 6 7 8 9 3 , 5 · · · 4 3 , 5 · · · 1 , 2 4 6 = 1 , 2 6 = 0 7 , 9 8 7 , 9 8 18/25

Action of Generators , , 1 2 3 4 5 6 7 8 9 3 , 5 · · · · · · 1 , 2 4 6 = 1 , 2 , 3 , 5 4 6 7 , 9 8 7 , 9 8 1 2 3 4 5 6 7 8 9 3 , 5 3 , 5 · · · · · · = 1 , 2 1 , 2 4 = 0 4 6 7 , 9 6 , 7 , 9 8 8 19/25

Restricts Naturally to Subalgebras of the Partition Algebra Brauer Algebra Temperley-Lieb (matchings) (planar matchings) 1 , 5 6 , 9 · · · 2 , 3 1 , 5 8 , 9 · · · 4 6 7 2 4 7 3 8 Motzkin Algebra Rook Brauer Algebra (planar partial matchings) (partial matchings) · · · 3 , 4 5 8 7 , 9 · · · 2 1 , 4 5 6 , 8 1 2 6 3 9 7 20/25

Restricts Naturally to Subalgebras of the Partition Algebra Symmetric Group Identity (permutations) (planar permutations) · · · · · · 1 2 5 8 1 2 3 4 5 6 7 8 9 3 4 9 6 7 Rook Monoid Planar Rook Monoid (partial permutations) (planar partial permutations) 3 4 5 7 1 2 3 5 6 9 · · · · · · 1 2 8 4 7 8 6 9 21/25

III. Insertion Bijections 21/25

X k , n ) 2 ( m � III. Insertion Bijections: dim( P k ( n )) = � ` n 1 2 3 4 5 6 0 1 1 , 3 [COSSZ’19] 5 , 2 , 3 6 2 4 @ A 5 , 6 1 , 4 1 2 3 4 5 6 [HL’04] [BH’19] ∗ [CDDSY’06] [COSSZ’19] 0 1 2 3 4 5 6 ⇣ ⌘ P = 0 1 2 3 4 5 6 ⇣ ⌘ Q = 22/25